The classical Alexander polynomial is an important topological invariant associated to a link in a 3-sphere (or, more generally 3-manifold). It was first introduced by Alexander [1]}. On the other hand, there is a \(p\) -adic version introduced by Hillman-Matei-Morishita [2]} which is widely regarded as the multi-variable topological analogue of the Iwasawa polynomial of \(\mathbb {Z}_p\) -extension. In this setting, one may define \(\mu \) , \(\lambda \)  and \(\nu \)  invariants, and these invariants give a formula for the asymptotic growth of the \(p\) -parts in the homology for the various finite covering spaces occuring in a chosen \(\mathbb {Z}_p\) -cover which is branched along \(\mathcal {L}\) . In this section, we recall the notion of the completed Alexander polynomial and associated Iwasawa invariants, and explain the relationship with \(p\) -homology classes occurring in a \(\mathbb {Z}_p\) -cover.
